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Higher-order Derivatives

The first derivative is the slope of a curve, and the second derivative is the slope of the slope, like acceleration. So what's the third derivative? (Fun fact: it's actually called jerk.)

Characteristics of f, f', f''

Let $$f$$ be a function such that $$f(0)=50.$$ If the above graph represents the graph of the first derivative of $$f$$ on the interval $$0<x<3,$$ which of the following is the correct description of $$f$$ on this interval?

For the function $$y=x^3+x^2-x-1,$$ determine the interval where $$y$$ is decreasing.

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Suppose $$f$$ is a function such that $$f(0)=33$$ and the first derivative of $$f$$ on the interval $$-2 < x < 2$$ is as shown in the above diagram. Which of the following is the correct description of the function $$f$$ on this interval?

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Suppose $$f$$ is a function defined on the closed interval $$-3 \le x \le 4$$ with $$f(0)=42,$$ such that the graph of $$f',$$ the derivative of $$f,$$ on the interval is as shown in the above diagram. On what intervals is $$f$$ increasing?

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Suppose $$f$$ is a function defined on the closed interval $$-3 \le x \le 4$$ with $$f(0)=3$$ such that the graph of $$f',$$ the derivative of $$f,$$ on the interval is as shown in the above diagram. Find the $$x$$-coordinates of the points of inflection of $$f.$$

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